Needs["VectorAnalysis`"]
Clear[field, r, T, F];
m = Transpose[
Transpose[JacobianMatrix[#, Spherical @@ #]]/
ScaleFactors[Spherical @@ #]] &@{r, T, F};
field[r_, T_, F_] =
Simplify[m.{3/(4 r^4) (3 Cos[T]^2 - 1),
3/(4 r^4) Sin[2 T], 0}];
StreamPlot[
Delete[
field @@ CoordinatesFromCartesian[{x, 0, z}, Spherical], 2],
{x, -3, 3}, {z, -3, 3}]